Show that the vectors $u_1 = (1/9,4/9,8/9), u_2=(8/9,-4/9,1/9), u_3=(-4/9,-7/9,4/9)$ form an orthonormal basis of $\mathbb{R}^3$. Find the projection of any vector $x=(\xi_1,\xi_2,\xi_3) \in \mathbb{R}^3$ onto the linear span of the vectors $u_1$ and $u_2$, and also the normal from any vector to that same linear span. Both, taking the projection and the normal, are linear operators. Write down their matrices in the usual basis of $\mathbb{R}^3$.

I understand and can easily prove that the vectors form an orthonormal basis. Past that, I am stumped.


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